How to return the name of the calling script from a Powershell Module?

Use PSCommandPath instead in your module:
Example test.psm1

function Get-Info{
    $MyInvocation.PSCommandPath
}

Example myTest.ps1

Import-Module C:\Users\moomin\Documents\test.psm1 -force
Get-Info

Output:

C:\Users\moomin\Documents\myTest.ps1

If you want only the name of the script that could be managed by doing

GCI $MyInvocation.PSCommandPath | Select -Expand Name

That would output:

myTest.ps1

I believe you could use the Get-PSCallStack cmdlet, which returns an array of stack frame objects. You can use this to identify the calling script down to the line of code.

Module: test.psm1

Function Get-Info {
    $callstack = Get-PSCallStack
    $callstack[1].Location
}

Output:

myTest.ps1: Line 2